Invariants
Base field: | $\F_{13}$ |
Dimension: | $2$ |
L-polynomial: | $( 1 - 4 x + 13 x^{2} )( 1 - 3 x + 13 x^{2} )$ |
$1 - 7 x + 38 x^{2} - 91 x^{3} + 169 x^{4}$ | |
Frobenius angles: | $\pm0.312832958189$, $\pm0.363422825076$ |
Angle rank: | $2$ (numerical) |
Jacobians: | $0$ |
Isomorphism classes: | 3 |
This isogeny class is not simple, primitive, ordinary, and not supersingular. It is principally polarizable.
Newton polygon
This isogeny class is ordinary.
$p$-rank: | $2$ |
Slopes: | $[0, 0, 1, 1]$ |
Point counts
Point counts of the abelian variety
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $110$ | $33660$ | $5239520$ | $823996800$ | $137389054550$ |
Point counts of the (virtual) curve
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $7$ | $197$ | $2380$ | $28849$ | $370027$ | $4819034$ | $62737591$ | $815785921$ | $10604807500$ | $137858870957$ |
Jacobians and polarizations
This isogeny class is principally polarizable, but does not contain a Jacobian.
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{13}$.
Endomorphism algebra over $\F_{13}$The isogeny class factors as 1.13.ae $\times$ 1.13.ad and its endomorphism algebra is a direct product of the endomorphism algebras for each isotypic factor. The endomorphism algebra for each factor is: |
Base change
This is a primitive isogeny class.